ANNIHILATING PROBABILITY MEASURES UNDER CONSTRAINTS
BERNHARD BURGSTALLER
Abstract. Let P be a probability measure and H ⊆ L∞ (P) be a linear subspace and
0 < c ≤ 1 ≤ C real constants. Then we give a relatively computable criterion whether or
not there exists a H-annihilating probability measure Q ∼ P equivalent to P with density
c ≤ dQ/dP ≤ C. In fact we also prove a version where L∞ (P) is replaced by C(K) for a
compact Hausdorff space K.
1. Introduction
In the theory of finance there exists an import theorem which is well-known under “the
fundamental theorem of asset pricing”. Mathematically spoken the following question is a
central building block for its solution. One has given a probability measure P, a subspace
H ⊆ L∞ (P), say, and the question is whether there exists an equivalent probability measure
R
Q ∼ P such that f dQ = 0 for all f ∈ H (Q is H-annihilating).
This question and its answers, in connection with stochastic processes, are well discussed
and a heap of papers deals with them (samples are [1, 2, 3, 4, 5, 6, 8, 9]). However, we think
a pure mathematical aspect of this building can be enriched, for we asked for H-annihilating,
equivalent probability measures with constraint on the density function dQ/dP. We obtained
Theorem 1.1 below.
R
For a integrable function f let I(f ) = f dP. If I(f ) ≤ 0, then let G(f ) be the unique
R
real number essinff ≤ δ ≤ 0 such that f ∨ δdP = 0.
Theorem 1.1. Let P be probability measure, H ⊆ L∞ (P) a linear subspace and 0 < c ≤ 1 ≤
C real constants. Then there exists a H-annihilating equivalent probability measure Q ∼ P
such that c ≤ dQ/dP ≤ C if and only if for all f ∈ H and all α ∈ [I(f ), esssupf ] one has
0 ≤ (C − 1)α + (c − 1)G(f − α) − CI(f ).
1991 Mathematics Subject Classification. 46E15, 46E27.
Key words and phrases. annihilating, probability measure, equivalent measure, compact space.
Institute of Mathematics, University of Münster, Einsteinstraße 62, 48149 Münster, Germany.
The author was supported by the Austrian Research Foundation (FWF) project S8308. This research
was done at the University of Linz in Austria.
1
2
BERNHARD BURGSTALLER
It is easy to check that the criterion is positive homogenous, i.e. the inequality is true for
f ∈ L∞ (P) and all α iff it is valid for λf (λ > 0) and all α. In consequence it is already
sufficient to check the criterion just for all f ∈ H ∩ S, where S is the sphere of the unit ball,
say.
The criterium is particularly useful if L∞ (P) has high dimension and H has low dimension.
If H is finite dimensional then H ∩ S is a [dim(H) − 1]-dimensional compact sphere. If the
above inequality is then numerically testable for one function f and one value α then it is
thinkable that the inequality is testable for all f ∈ H ∩ S and the corresponding α’s up to
a certain precision.
In the following example we consider the case when the probability space Ω is finite. In
this case the above theorem can ensure a root for real-valued matrix A such that the root
satisfies certain constraints.
Example 1.2. Let A be a real valued m × n-matrix. Let a ≤ b be positive real constants.
P
Does there exist a solution ξ ∈ Rn of the equality Aξ = 0 such that 1 = |ξ| := ni=1 ξi
and a ≤ ξi ≤ b for all i = 1, ..., n? Let a1 , ..., am be the rows of A. A solution ξ of the
equality Aξ = 0 exists if and only if < ai , ξ >= 0 for all i = 1, ..., m. Equivalent is, that
P
< f, ξ >= ni=1 fi ξi = 0 for all f of the vector space H := span{a1 , ..., am }. Regarding
Ω = {1, ..., n} as a measurable space we have to find a H-annihilating probability measure
ξ.
Let P({i}) = n1 . Then the constraint a ≤ ξi ≤ b is equivalent to the constraint an ≤
ξn ≤ bn where dξ/dP = ξn. Hence the desired ξ exists if and only if the condition given in
Theorem 1.1 holds.
We also obtain a result like Theorem 1.1 for the space C(K), where K is a compact
Hausdorff space (Theorem 2.3). Moreover we give criterions for H-annihilating, equivalent
probability measures Q ∼ P and less restricting, or lets say, other constraints on the density
function dQ/dP in section 3. But all these criterions contain several quantors and are by
far less computable then the criterion given in Theorem 1.1.
2. Annihilation in C(K)
In this section we prove the C(K)-version of Theorem 1.1. The version for L∞ (P) can
then be easily deduced from the C(K)-version due to Gelfands representation theorem. See
also the discussion in the next section.
In what follows K denotes a compact Hausdorff space. In this paper all function spaces
are real-valued. A state on C(K) is a positive linear functional ϕ : C(K) → R with ϕ(1) = 1.
ANNIHILATING PROBABILITY MEASURES UNDER CONSTRAINTS
3
Two functionals ϕ, ψ ∈ C(K)∗ satisfy per def. the relation ϕ ≤ ψ if ϕ(f ) ≤ ψ(f ) for all
positive functions f ∈ C(K). The maximum and minimum of f ∈ C(K) is denoted by
max(f )/ min(f ); the unit of C(K) by 1.
Proposition 2.1. Let K be a compact Hausdorff space, ϕ a state on C(K), H ⊆ C(K) a
linear subspace and c ≤ 1 ≤ C reals. Then there exists a H-annihilating linear functional ψ
on C(K) with ψ(1) = 1, cϕ ≤ ψ ≤ Cϕ if and only if for all x ∈ H
(1)
ϕ(x) ≤ inf (C − 1) max[x + y − ϕ(x + y)1] + (1 − c) max[y − ϕ(y)1].
y∈C(K)
Proof. We start with some preparations. Let E = ker(ϕ) and
P : C(K) → E : P x = x − ϕ(x)1.
Then we write C(K) as direct sum C(K) = E ⊕ R1 using the projection P , i.e. x =
P x + ϕ(x)1 in this coordinate system. Abbreviate
m : C(K) → R : m(x) = − min(x).
If we replace y by −y in the inequality (1), then it reads
(2)
ϕ(x) ≤ inf (C − 1)m(−P (x − y)) + (1 − c)m(P (y)).
y
The only if part. Now assume that we have given the annihilating functional ψ with the
stated properties. Note that clearly x + m(x)1 ≥ 0 for any x ∈ C(K). The following
conditions (3) and (4) are seen to be equivalent by using a few elementary manipulations
(note that ϕ(1) = ψ(1) = 1 and ϕ(x) = 0 for all x ∈ E).
(3)
∀x ∈ E : cϕ(x + m(x)1) ≤ ψ(x + m(x)1) ≤ Cϕ(x + m(x)1)
(4)
∀x ∈ E : −ψ(x) ≤ (1 − c)m(x) and − ψ(x) ≤ (C − 1)m(−x)
Now due to our assumptions for any x ∈ H we have 0 = ψ(P x) + ϕ(x). Hence we get the
desired result, i.e. for any y ∈ C(K)
ϕ(x) = −ψ(P x) = −ψ(P (x − y)) − ψ(P y)
≤ (C − 1)m(−P (x − y)) + (1 − c)m(P (y)).
The if part. Note that
x 7→ α(x) := (C − 1)m(−x)
and
x 7→ β(x) := (1 − c)m(x)
4
BERNHARD BURGSTALLER
are positive sublinear functionals on E, where the positivity follows from max(x) ≥ ϕ(x) =
0. Thus their ”convex hull”
(5)
n(x) := inf{ α(x − y) + β(y) | y ∈ E }
(x ∈ E)
is also sublinear on E. Now we have a well-defined function
(6)
l : P (H) → R : l(P x) = ϕ(x),
since P x = 0 gives x = ϕ(x)1 ∈ H and ϕ(x) 6= 0 would then contradict the inequality (2)
by setting x = y = 1. We use the inequality (1) and get for all x ∈ H
l(P (x)) = ϕ(x) ≤ n(P x).
Via Hahn-Banach we extend l to a linear map ˆl : E → R with ˆl ≤ n. Now the functional
(7)
ψ : C(K) → R : ψ(y) = ϕ(y) − ˆl(P y)
obviously annihilates H. For x ∈ E we get
−ψ(x) = ˆl(x) ≤ n(x) ≤ α(x), β(x).
This is exactly (4) and we obtain (3). Since c ≤ 1 ≤ C, for real λ ≥ 0 we can add to (3)
the inequality cϕ(λ1) ≤ ψ(λ1) ≤ Cϕ(λ1). If we put λ2 = m(x) + λ we obtain
cϕ(x + λ2 1) ≤ ψ(x + λ2 1) ≤ Cϕ(x + λ2 1).
Since x ∈ E and λ2 ≥ m(x) were arbitrary, and each positive y ∈ C(K) has the representation y = x + λ2 1 with x ∈ E and λ2 ≥ m(x), we have shown that cϕ ≤ ψ ≤ Cϕ. The
boundedness of ψ clearly follows from the inequality 0 ≤ ψ − cϕ ∈ C(K)∗ . ¤
Proposition 2.2. Let K be a compact Hausdorff space, ϕ a state on C(K), 0 ≤ c, C reals
and x ∈ C(K). Then the infimum
(8)
I = inf C max[x + y − ϕ(x + y)1] + c max[y − ϕ(y)1]
y∈C(K)
is attained by y = −((x − α1) ∨ β) for some reals α, β ∈ R with ϕ(x) ≤ α ≤ max(x) and
β ≤ 0 such that ϕ(y) = 0.
Moreover its value is
¯
o
n
¯
I = min Cα − cβ ¯ϕ(x) ≤ α ≤ max(x), β ≤ 0, ϕ((x − α1) ∨ β) = 0
−Cϕ(x).
ANNIHILATING PROBABILITY MEASURES UNDER CONSTRAINTS
5
Proof. For the following computations we will need also discontinuous functions and therefore actually we will consider the larger space `∞ (K) (all bounded functions on K) instead
of C(K). For this reason we extend ϕ via Hahn-Banach to a state ϕ̃ ∈ `∞ (K) (ϕ̃ is a state
since ϕ̃(1) = ϕ(1) = kϕk = kϕ̃k, what already characterizes positivity, see e.g. [7, 4.3.2]).
Assume that we have shown the lemma for the case that C(K) is replaced by `∞ (K) and ϕ
is replaced by ϕ̃. Then it is clear that our aimed C(K)-version holds too, since the infimum
in (8) is already attained by the continuous function y = −[(x − α1) ∨ β] ∈ C(K).
So we shall consider `∞ (K) with a given state ϕ on it. We put
E := ker(ϕ) = { y − ϕ(y)1 | y ∈ `∞ (K) }.
We decompose now all vectors in `∞ (K) into its positive and negative parts where for the
positive part we use capital letters and for the negative part we use lower case letters.
Before we proceed with the proof, we notice the following facts which we will use in
the sequel several times. First of all, the transition from C(K) to `∞ (K) imposes the
replacement of min and max by inf and sup (where sup(f ) := supω∈K f (ω), f ∈ `∞ (K)).
Next, if f ∈ `∞ (K), sup(f ) ≥ 0 and A ⊆ K, then we have
(9)
sup(f ) = sup(f 1A ) ∨ sup(f 1K\A ).
So this formula is especially applicable for functions Y − y ∈ E (Y, y ≥ 0, Y y = 0) since we
have
0 = ϕ(Y − y) ≤ sup(Y − y).
Thus, for example, sup(Y − y) = sup(Y ) ∨ sup(y) = sup(Y ). Good to know is also the easy
identity
(10)
(f g) ∨ 0 = f (g ∨ 0)
for a positive function f and an arbitrary function g, and hence we will often omit the
brackets. Finally, we use the notion
S(f ) = 1{ ω∈K | f (ω)6=0 }
for the carrier of f .
Going back to the proof, we replace the expression x − ϕ(x)1 ∈ E by X − x ∈ E in (8)
(X, x ≥ 0, xX = 0, the x in x − ϕ(x)1 is not identic with the x in X − x (!)) to get good
notations. At the end of the proof we will abandon this substitution. Then, in order to get
the value (8), we have to look for Y − y ∈ E which minimizes the value
(11)
C sup(X − x + Y − y) + c sup(Y )
6
BERNHARD BURGSTALLER
(Y, y ≥ 0, Y y = 0) and notice that all numbers occurring in (11) are positive.
Optimization step 1: Let Y − y ∈ E be given. We are going to improve (11) by several
manipulations on Y − y. Put
z = S(X)y.
We obtain
sup(X − x + Y − y) = sup S(X + Y )(X − x + Y − y)
= sup(X − x + Y − z).
Optimization step 2: We put
α = 0 ∨ sup S(z)(X − z),
w = S(z)(X − α1) ∨ 0.
Observing that
(12)
sup S(z)(X − w) = sup[S(z)(X − α1) − S(z)(X − α1) ∨ 0 + αS(z)] = α
we proceed (we use here (9))
0 ≤ sup(X − x + Y − z)
= sup S(z)(X − z) ∨ sup(1 − S(z))(X − x + Y )
(13)
= sup S(z)(X − w) ∨ sup(1 − S(z))(X − x + Y )
(14)
= sup(X − x + Y − w).
Note that w ≤ z ≤ y since
S(z)X − z = S(z)(X − z) ≤ (sup S(z)(X − z))1 ≤ α1.
We put W = ϕ(w)ϕ(y)−1 Y and thus W ≤ Y and W − w ∈ E since ϕ(W ) = ϕ(w). It is
then advantageous to replace Y − y by W − w,
(15)
sup(X − x + Y − y) = sup(X − x + Y − w)
(16)
≥ sup(X − x + W − w) =: γ ≥ 0.
Optimization step 3: We put
v = S(z)(X − γ1) ∨ 0.
One can calculate (by considering the two cases X(ω) ≥ γ, X(ω) < γ, ω ∈ K, say)
(17)
sup S(z)(X − v) = γ ∧ sup S(z)X.
ANNIHILATING PROBABILITY MEASURES UNDER CONSTRAINTS
7
Note that sup(X − x + Y − w) ≥ α due to (12), (13) and (14). Combining this with (15)
and (16) yields α ≥ γ.
Thus v ≤ w and we put V = ϕ(v)ϕ(w)−1 W . It is then advantageous to replace W − w
by V − v ∈ E. Indeed, since v ≤ w ≤ y we have
(18)
S(z)v = v,
S(z)w = w,
and
γ = sup(X − x + W + w) ≥ sup((1 − S(z))(X − x + W )
≥ sup((1 − S(z))(X − x + V )
yields therefore with (15), (16) and (17)
sup(X − x + Y − y)
(19)
≥ γ ≥ sup S(z)(X − v) ∨ sup(1 − S(z))(X − x + V )
= sup(X − x + V − v).
Since (note (18) and (16))
(1 − S(z))(X − x) ≤ (1 − S(z))(X − x + W − w) ≤ γ1
we obtain (note also (10) and S(z)x = 0)
(20)
(X − x − γ1) ∨ 0 = S(z)(X − x − γ1) ∨ 0 = v.
If γ ≤ sup S(z)X then, due to (19) and (17), we obtain the equality
(21)
γ = sup(X − x + V − v).
Thereby v is defined by (20) and
(22)
γ ≤ sup(X).
The same facts we obtain for the contrary case γ > sup S(z)X, if we just replace γ
by γ2 := sup(X). Indeed in this case we have v = V = 0 and v = 0 is expressed by
v = (X − x − γ2 1) ∨ 0 like in (20), and we get γ2 = sup(X − x) = sup(X − x + V − v) like
in (21).
Optimization step 4: It became clear that we can restrict us to the following case in the
optimization problem (11): We have a scalar
(23)
0 ≤ α ≤ sup(X)
8
BERNHARD BURGSTALLER
such that
(24)
y = (X − x − α1) ∨ 0,
(25)
sup(X − x + Y − y) = α
(see (20), (21) and (22)). We are going to optimize Y . We have, inserting (24),
(26)
0 ≥ X − x + Y − y − α1 = (X − x − α1) ∧ 0 + Y.
Therefore we obtain the following necessary condition on Y :
(27)
0 ≤ Y ≤ −((X − x − α1) ∧ 0).
On the other hand, if some Y satisfies (27), then obviously Y y = 0, and equality (25) holds
further, since using the equality in (26) and (24) and α ≤ sup(X) we get
0 ≥ sup(X − x + Y − y − α1) ≥ sup(X − x − α1 − y) = 0.
Thus we can feel free to choose any Y satisfying (27) without touching the left summand in
(11) and we just have to optimize sup(Y ).
Now, for the only further necessary condition on Y (beside (27)) is ϕ(Y ) = ϕ(y), it is
clear that the best choice is to simply cut the righthanded function in (27) function, i.e. to
choose a β ≤ 0 and put
(28)
Y = (−β) ∧ (−((X − x − α1) ∧ 0)) = −((X − x − α1) ∧ 0 ∨ β).
To be precise such β exists since for arbitrary z ∈ A the function R → R : β 7→ ϕ(z ∨ β) is
continuous and hence the necessary condition
(29)
0 = ϕ(y − Y ) = ϕ((X − x − α1) ∨ β)
has a solution for some β ≤ 0 (recall (23)). So
(30)
Y − y = −((X − x − α1) ∨ β)
is now of the desired form and (11) becomes Cα − cβ (due to (25) and (28); alternatively
one simply inserts (30) in (11)).
Final step: Therefore (11) attains its infimum by (recall (29))
¯
o
n
¯
inf Cα − cβ ¯0 ≤ α ≤ sup(X − x), β ≤ 0, ϕ((X − x − α1) ∨ β) = 0 .
It is clear that the infimum is attained here, as (α, β) can be chosen from a compact set.
More detailed (α, β) ∈ [0, sup(X − x)] × [inf(X − x), 0] ∩ ker(f ) where f is the continuous
function f : R × R → R : f (α, β) = ϕ((x − α1) ∨ β).
ANNIHILATING PROBABILITY MEASURES UNDER CONSTRAINTS
9
Last but not least we finish the proof by abandoning the ”substitution” x−ϕ(x)1 ≡ X −x
of the beginning and easily obtain the announced value for I. ¤
The combination of propositions 2.1 and 2.2 yields
Theorem 2.3. Let K be a compact Hausdorff space, ϕ a state on C(K), H ⊆ C(K) a linear
subspace and c ≤ 1 ≤ C reals. Then there exists a H-annihilating functional ψ ∈ C(K)∗
with cϕ ≤ ψ ≤ Cϕ and ψ(1) = 1, if and only if
(31)
∀f ∈ H : ϕ(f ) ≤ C −1 min(α(C − 1) + β(c − 1)),
where the minimum is taken over all α, β ∈ R with ϕ(f ) ≤ α ≤ sup(f ) and β ≤ 0 such that
ϕ((f − α) ∨ β) = 0.
The next corollary is the C(K)-version of Theorem 1.1. Let us recall some definitions
before. A functional ϕ ∈ C(K)∗ is said to be normal if for all nets in the unit ball (fi )i∈I ⊆
BC(K) converging to zero (∀x ∈ K : fi (x) → 0) we have ϕ(fi ) → 0.
A state ϕ is faithful if ϕ(f ) > 0 for all f > 0.
Corollary 2.4. Let ϕ be a faithful normal state on C(K) and 0 < c ≤ 1 ≤ C. Then
there exists a H-annihilating faithful normal state ψ with cϕ ≤ ψ ≤ Cϕ iff criterion (31) is
fulfilled.
3. Annihilation in L∞ (P)
We review now our last results under the light of probability measures. Consider a
probability space (Ω, F, P). Then we have an isometric isomorphism L∞ (Ω, F, P) ∼
= C(K)
via Gelfand’s transformation and where K is stonean ([10, I.4.4] or [7, 5.2.1]).
Alternatively, all proofs of the previous section would also go through for L∞ (P) if one
just replaces max(f ) by ess sup(f ). Thereby one can skip a transition like from C(K) to
`∞ (K) in the proof of Proposition 2.2, for we have all necessary projections in L∞ (P) and
so all operations, which have been needed, act in L∞ (P) anyway. So actually the Gelfand
representation is not really necessary.
Now if we consider the functional
Z
∞
(32)
ϕ : L (P) → R : ϕ(f ) = f dP
and apply Theorem 2.3 for 0 < c ≤ 1 ≤ C, then we directly obtain our main result Theorem
1.1. This is pretty clear: A functional ψ with cϕ ≤ ψ ≤ Cϕ and ψ(1Ω ) = 1 defines an
10
BERNHARD BURGSTALLER
equivalent probability measure Q ∼ P via
(33)
Q(A) = ψ(1A )
A ∈ F.
And the estimate cϕ ≤ ψ ≤ Cϕ is equivalent to c ≤ dQ/dP ≤ C.
In the next result we weaken the constraint on Q to the one sided estimate dQ/dP ≤ C.
But the criterion is abruptly more difficult to decide for it contains more quantors.
Corollary 3.1. Let P be probability measure, H ⊆ L∞ (P) a linear subspace and 1 ≤ C.
Then there exists a H-annihilating, equivalent probability measure Q ∼ P with density
dQ/dP ≤ C, if and only if for each measurable set A with P(A) > 0 there exists a smaller
measurable set B ⊆ A with P(B) > 0 and a real γ > 0 such that
Z
1
∀f ∈ H ⊕ [1B − γ1] : f dP ≤ min (α(C − 1) − β),
C
R
where the minimum is taken over all α, β ∈ R with f dP ≤ α ≤ sup f and β ≤ 0 such that
Z
(f − α) ∨ βdP = 0.
This can be proved by using an argument of Kreps [8] and Yan [11]. We skip the proof.
In the next corollary we have no restriction on the density function dQ/dP. This corollary
should just point out, that the previous versions under constraint, are ”dense” in a certain
sense within the cases where we have no restriction on dQ/dP.
Corollary 3.2. Let ε > 0. Then the following conditions are equivalent.
(1) There exists a H-annihilating probability measure Q ∼ P.
(2) There exists a probability measure R ∼ P and reals 0 < c ≤ 1 ≤ C such that
kdR/dP − 1kL1 (P) ≤ ε and the criterion of Theorem 1.1 is satisfied for R (instead of
P).
(3) For all measurable sets A ∈ F with P(A) > 0 there exist a P-absolute continuous
probability measure R with R(A) > 0 and reals 0 < c ≤ 1 ≤ C such that the criterion
of Theorem 1.1 is satisfied for R (instead of P).
We omit its proof.
In the next Proposition we give a necessary and sufficient condition for the existence of
an annihilating probability measure Q under the constraint dQ/dP ∈ Lq (P). However, one
easily realizes that the condition is in the best case good to falsify the existence of Q.
Proposition 3.3. Let P be a probability space, H ⊆ L∞ (P) a linear subspace and 0 < c ≤
1 ≤ C, 1 < q ≤ ∞ reals. We define p by 1/p + 1/q = 1. Then there exists a H-annihilating,
equivalent probability measure Q ∼ P with the following constraints on the density function.
ANNIHILATING PROBABILITY MEASURES UNDER CONSTRAINTS
11
(1)
dQ/dP ∈ Lq (P)
c ≤ dQ/dP,
if and only if there exists a real D ≥ 0 such that
∀x ∈ H∀y ∈ L∞ (P)
Z
Z
Z
°
°
³
´
°
°
xdP ≤ D°x + y − (x + y)dP1°
+ (1 − c) sup y − ydP1 .
P
Lp ( )
(2)
dQ/dP ∈ Lq (P),
if and only if for each nonzero projection f ∈ L∞ (P) there exist a nonzero projection
e ≤ f and reals γ > 0, D ≥ 0 such that ∀x ∈ H ⊕ [e − γ1]∀y ∈ L∞ (P)
Z
Z
Z
°
°
³
´
°
°
+ sup y − ydP1 .
xdP ≤ D°x + y − (x + y)dP1°
P
Lp ( )
We omit the proof. It is similar to the proof of Proposition 2.1 on the one hand, and by
using the method of Kreps and Yan on the other hand.
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12
BERNHARD BURGSTALLER
E-mail address: [email protected]